KOMATSU, TAKASHI

TAKASHIMA, KEIZO

Komatsu, Takashi

Takashima, Keizo

Osaka University Knowledge Archive

On the existence of intersectional local time except on zero capacity set

Institutional Repositories DataBase (IRDB)

Osaka City University Repository

Komatsu, T. and Takashima, K.
Osaka J. Math.
21 (1984), 913-929
ON THE EXISTENCE OF INTERSECTIONAL LOCAL
TIME EXCEPT ON ZERO CAPACITY SET
Dedicated to the memory of Professor Takehiko Miyata
TAKASHI KOMATSU AND KEIZO TAKASHIMA
(Received December 2, 1983)
0. Introduction
Let W be the space of /^-valued continuous functions on [0, 1], where
We shall consider the functionals on W
o
(0.1) ψ(α, w) = *=«- f
4 Jo J
which may take infinite value. These functionals play an important role in
the investigation of properties of function w: the finiteness of ψ(α, w) implies
that the Hausdorίf dimension of range {w(ί)\ O^ί^l} is no less than a (cf.
Taylor [9]). Let Q be the Wiener measure on W. Since ψ(α, w) is finite
^-almost surely for any α<2, the Hausdorίf dimension of {w(t); O^ί ^  l}
is no less than 2 ^-almost surely.
Next, let a tend to 2. Though the mean of ψ(#, •) with respect to Q
diverges to infinity, the functional
(0.2) Ψ.(w) = ψ{2-2-», w)-2n
converges ^-almost surely. In case d=2y Varadhan studied this limit func-
tional in connection with the quantum field theory and proved its existence
(cf. Appendix to Symanzik [8]).
Recently Fukushima [1] showed that various famous properties of sample
paths such as Lόvy's Holder continuity hold not only Q-almost surely but
also quasί-every where
 9 i.e. except on a set of zero capacity with respect to the
Ornstein-Uhlenbeck process on W. On the other hand, Kόno [4] and [5]
proved that if d^,4, then sample paths are recurrent with positive capacity.
Therefore 'quasi-everywhere* is strictly finer than ^-almost everywhere'.
The purpose of this paper is to show that ^]r(a, w) is finite quasi-every where
for any α<2 and that lim Ψ
n
(w)=Ψ(w) exists quasi-everywhere. The former
result implies the theorem in Komatsu and Takashima [3]: the Hausdorff
914 T. KOMAΊSU AND K. TAKASHIMA
dimension of range {w(t); O^ί^l} is 2 quasi-everywhere. Let (Ω, 3, P,
J?
τ
( )) be the Ornstein-Uhlenbeck process on W. Since a Borel subset A
of W has zero capacity if and only if
P[X£ )&A foranyτ] = l
(cf. Fukushima [1], Kusuoka [6]), it is sufficient to prove the continuity of -fy(a>
X
τ
( )) in T and the uniform convergence of ψ
n
(X
τ
( )) in T.
In case d=2, the limit functional lίmψ
n
(w)=ψ(w) is formally expressed
by
= T I' Γ2 Jo Jo
(C is an infinite constant), which is similar to the inter sectional local time con-
sidered in Wolpert [11]. Westwater [10] investigated a similar functional in
connection with the study of long polymer chains in R3. Finally, we shall
mention the relative result of Shigekawa [7]: the 1 -dimensional Brownian
local time exists quasi-everywhere.
The authors wish to express their hearty gratitude to Prof. M. Fukushima
and to Prof. S. Kusuoka for their suggestion and advice.
1. Singular Wiener functional
Let Wbe the Banach space of all /^-valued continuous functions w=(w(t))
on [0, 1] satisfying zϋ(0)=0; cϊ^, the usual Borel field and Q, the Wiener measure
on (V, W). Set, for
(1.1) /.(*, x) = -— {(|*
z — a.
Considering that Δ/
ε
(α, x) — > (d—ά) \ x \ ~* as 6 I 0, we shall define
(1.2) 
β^
(«, w) = J J i- Δ/8(α, w(ί, ί))
where Δ denotes the Laplacian and w(sy ί)=w(t)—w(s).
Set Qj=dldxj and 3^(9!, •••, dd). From the Ito formula
.(«, 0) = Γ (/8(α, «<*, 1))-Γ 9/ε(^, w(*,
Jθ *^5
Using the Fubini type theorem for the product ds dw(i) (cf. Ikeda and Wata-
nabe [2] Chap. II Sec. 4 Lemma 4.1), we have
a, 0) = /.(«, ^  1))
iNTFRStCTIONAL LOCAL TlME 915
Let g
ζ
(a, x) be the isotropic function satisfying
(α, Λ?)=/
e
(α, x) and ge(a,Q) = Q.
Then g9(a, x) is given by
(1.3) g,(a, x) = " r1-' 2/,(«, «|) ««-' <fo dr, |f
Let #' denote the transposed vector of xy x y=x'y, the inner product of column
vectors x and j>, and 9'9=(9I 9y): t/xrf-matrix. Define
Γ h(t, w) άw(ί) = LMim Σ h (ΐ-9 w] wί-^-, -*-Jo »->« ι=ι \ n J \ n n
for a process h(t, w) adapted to σ-fields σ(w(u)\ t^u^l). Then we see that
j/.(α, «<*, 1)) * = g,(
a> w(Q, 1))-J^ dg,(a, w(ί, 1)) rfw(ί) ,
Γ 3/t(α, a ίί, ί)) ώ = 9g,(a, w(0, ί))-(' (ίί«;(ί))' 3Jo Jo
Therefore we have
(1.4) ψ-,(«, «>)+/,(«, 0) = g,(a, w(0, 1))
-t 3^ «(α, «>(*, 1)) rfa>(ί)-( 8^ , (α, α>(0,
Jo Jo
+ J J
a
-
e
Define, for α
(1.5) &(α,
= 2|«|2 ί l a l2-*-! 4+^-tf
(4-α) (2+d-a) I 2-α 2d
It is easy to show that, for \v\ ^2,
where v=(vly •••, vd)eZi, |ι»|=ι»,-| ----- \-vd and
916 T. KOMATSU AND K. TAKASHIMA
Let ψ(#, w) be the functional defined by (0,1). Since d 2^2,
ψs(a, w) -> ι/r(α, w) for all w as £ J O .
From (1.4) we have
(1.6) ψ (α, w)- -±- =
 go(a, w(0, 1))
-Γ dgo(a, »(*,Jo o
Q'.Qgι(a, w(s, f))dw(t) a.e.
The following theorem is proved in Section 2.
Theorem 1. (ψ
Λ
(α;)} converge for almost all w^W and the limit func*
tίonal lim Ψ
n
(w)=Ψ(w) satisfies
(1.7) Ψ(w) - £MO, 1))- Γ θ^ίwίί, 1)) rf^)- f 9 (^0,
Jo Jo
+ J a.e.,
(1.8) )^ = |^ |
Let (Ω, £F, P) be a probability space; B(dτXdt)=(B*(dτχdt)), a rf-dimen-
sional two-parameter white noise and BQ(t)=(BQ(t))y a J-dimensional Brownian
motion independent of B(dτ x dt) satisfying 50(0)=0. Define
(1.9) Xί(ί) - ^-τ/2 {SS(ί)+ (T ^ 2 B?(Arχ [0, ί])} .
The process X
r
= (X !,.(•)) is called the Ornstein-Uhlenbeck process on W.
Fix T and σ. Then the process ίAΛΛ->^Γ
τ
(ί) is a rf-dimensional Brownian motion
and
We shall prove the following theorems.
Theorem 2. For α/ry 0< # <2,
P[-vJr(tf, Jί
τ
) ώ continuous in τ] — 1 .
From the theorem we see that ιjr(α, w)<oo quasί-every where, ί.^ . except
INTERSECΠONAL LOCAL TIME 917
on a zero capacity set. Especially Ψ
n
(w)<oo quasi-everywhere.
Theorem 3. The sequence {Ψ
Λ
(«#)} converges quasi-eveiy where. Let
ψ(w)=lim Ψ
n
(w) quasi-everywhere. Then
P[Ψ(X
τ
)is continuous in τ] = 1 .
2. Elementary inequalities
Fix 0<α<2 and set 2/3=2— a. We shall consider the functions
/O 1 \ /^* / Λ / \ / \
Let k,=β-\€ίβ—(2€)2β) and
φ,(u) =* ^
Then we have
/(?(*)= ΓίΐφMK"-1Jo \Jo
(2.2)
8/7.(*) = x, £ φ
β
( I * I β) u"-1 du+~ k, x, ,
θiθyGΛ*) ==*,*, |*| -J$.(|*|)
;.-rf *, ^ 1*1 -
2) Γ φ.( I * I «) M"-1 du+\ 8{j k, .Jo a
In this and the following sections, we shall use the convenient practice
of letting c 's stand for unimportant positive constants which may change from
line to line.
Lemma 2.1. There is a constant C independent of £ such that
(2.3) |9vG
e
(*)|^C|*|2-'v'£2β,
(2.4) |9vGf(*+;y)-8vσ^
for any x, y, 0<£< 1 and \v\<*2.
Proof. Since 0^φ,(u)^~k^c ε2β9 inequality (2.3) follows from (2.2).
Set G((x)=Gz(x)— kζ\x\2/2d. It suffices for the proof of (2.4) to show that
(2.5) |9VG;(*+J)-9VG^
We see that 0^φt(ttj'Sέr*(a*)β, for φ,(u)^62β and
βφ^u) = ((ιf4 ^β—&)~((ιf+4^e^(&^^ .
Therefore we have
918 T. KOMATSU AND K. TAKASHIMA
It is easy to see that
9,- 9, 9, G',(χ) = h0(x)\x\$\x\
+*,(*) ΦK I * I )+*,(*) Γ Φ«( I * I «) «"-' Λ ,Jo
where A0, /^ and A2
 are
 homogeneous functions with index — 1. Since
we have
Now, suppose that | x \ <2 |y | . Then, for | v \ ^2,
1 8vσ;(*+y)-8Ό;(*) ι ^  1 8vσ;(*+y) ι + 1 Q^GW \
Suppose that \x\ ^ 2\y\. Then, for \v\^
1 8vG;(*+y)-8vGί(*) I = I Γ 99vG8/(Jo
These prove (2.5). q.e.d.
Next, define
(2.6) G\x) =
 go(2-2β, x)-gJ2-β, x)
= \x\2β~l (l I Γ ((\x\ruY-lV u"~l du] rdr .Jo \Jo /
Then we have
(2.7)
jG\x) = x, β-1 (( I * I «)p- 1)2 M1*-1 </M ,
•'f((|*|«)'-l)»*-lΛ.
Jo
Lemma 2.2. There is a constant C independent of β such that
(2.8) IβΌ^I^C/sμi'-
/or any x and \ v \ ^ 2 and that
INΊERSECTIONAL LOCAL TIME 919
(2.9) 1 8»G'(*)-9VGOO I ^  Cβ ( \ x" \ x \ -*-? \ y\ ~2 1 + 1 log -
X(l+|log|*| I) (1+ \log\y\ |) (1+(|*| V\y$2)
for any x, y and \ v |.=2, where Λ?V denotes
(a1)*! (**)v
Proof. By the inequality
(2.8) is easily proved from (2.6) and (2.7). For example, in case \v\ = 1,
1 6, G'(«) I ^ -f- 1 * I Γ (log( I x \ M))2 (1+|*| V)2 u'-1 du
4 Jo
We see that
log —
β log(«/w) \β-\nf-
Then it is easy to prove (2.9) from (2.7). For example,
|v|)2). q.e.d.
Proof of Theorem 1. Note that
From (2.8) we have, for | v \ <^2,
920 TV KOMATSU AND K. TAKASHIMA
Let Ψ(w) denote the right hand side of (1.7). From (0.2), (1.6) and the above
inequality, it is easily proved that
From
we know that Ψ
n
(w)->Ψ(w) a.e. as w->oo. q.e.d.
3. Preliminary estimates
Let Gβ(x) be the function defined by (2.6). From (1.6) we see that
(3.1) ^(2-2/3, Jf
τ
)~ψ(2-/3, -£•)+ - = WTT(Q, 1))
-f dG?(XT(s, l))dxj*)-r\l 9Gt(XJp, t)) dXτ(t)Jo Jo
+ j I ^XT(s)'d'dG%XT(s, t)) dXτ(t) a.e. (P) ,
where XT(s, t)=Xτ(t)-XT(s). Hence, for any 0^τ, σ^ 1,
-2A X
τ
)-ψ(2-β, X
τ
)-^2-2β, X
a
)+ψ(2-β, X
a
)\
+ 1 Γ {dG*(XT(s, 1)) <tX4t)-QG'(XJtt,Jo
+ Γ {3GP(^ T(0, ί)) ^ τ(ί)-3^β(^σ(0, ί))Jo
*<'
-dxj(s) WG*(XJ(s, 0) <ZΛΓ
β
(ί)} I
= I G WO, 1))-G 0^(0, 1)) I +3,+^ .
Let/>>2. Using the Burkholder inequality and (1.10), we have
(3.2)
Jo
+2(1 -«-"-'"») dG\XT(s, 1)) 9'G'(XJ& 1))}
+ί E| f {|9G^T(0, ί))-8G"(^ 0, ί))l2Jo
, ί))}
INTERSECTION AL LOCAL TIME 921
+c th E|8G^T(0, f))|> Λ .
Similarly we have
c- P E| P {Q'QGP(X4t, ί)) dX#)-d'dG*(Xj(s, ί))
Jθ Js
E| ' Q'
From the Burkholder inequality we see that
E I P {3'9G"(^
τ
(ί, ί)) dXM-Q'dG^Xά t
J s
+2(1 -
β
-ι
τ
-ι/») QdjG'(XJι, 0) d'djG^XA*, t))}
E I CWO, fb-ffΌ^XJp, t)) |
|V |=2 Jo
+c. (thinly72 2 P"f £ 1 9VG"(*T(0, ί)) I ' Λ ,\ 4 / ιvι=2 Jo
and
£ I Γ 9'9G^T(^, 0) ΛX W I p£c Σ I1"' ^  I PGUXJQ, t)) \>dt.Js |V|=2 Jo
Combining these inequalities, we have
(3.3) E I B2 1 *£c* Σ E I Γ 1 9VG^T(0, 0)-9vG^σ(0, i)) \2dt\ >'2|V |=2 Jθ
72
 , ί)) I' Λ .
For 0<:τ, σ^l, set
(3.4)
 α
Let β ί^) and B2(i) be independent rf-dimensional Brownian motions defined
on (ίϊ, £F, P) satisfying J31(Q)=JB2(0)=0. From (1.10) the law of the process
is equal to that of the process
t ΛΛM (B^bή+a B2(bt\ B^bή—a B2(bt)) .
922 T. KOMATSU AND K. TAKASHIMA
Therefore
= E I G^B1
f sup
By a similar argument we have the following lemma from (3.2) and (3.3).
Lemma 3.1. For p>2, it holds thai
(3.5) E I ψ(2-2£, χ
τ
)-ψ(2-β, X
τ
)-γ(2-2β, XJ+γ(2-β, X,) \ *
E \dvGp(B1(t))\f dt
+e
Jo
|V|=2
Fix 0<α<2 and let
 :
G,(Λ?) be the function defined by (2.1). Replace the
function Gβ(x) by the function G^(x) in the above arguments. Then we have
the following estimate, which is much simpler than (3.5).
Lemma 3.2. For fixed 0<α<2 andρ>2, it holds that
(3.6) EI ψt(αf ^τ)-ψ2ε(α, ^ )-ψt(α, XJ+Y»(<x, X.) I'
^ α*Σ sup|V|=1.2 /^l
sup
4. Moment inequalities
Let 2—a=2β>0. Define a function ζp(a), 0^α<l, by
(4.1) ζp(a) =\ap *
Lemma 4.1. There is a constant C independent of £ such that
(4.2) FJifc.f/v. Y\—ik.f/y Y\ — ^ (rv Y\Λ-^(n Y\\P
O^r, σ^l.
Proof. From (2.3) we see that, for | v\ ^2,
INTERSECT IONAL LOCAL TIME 923
Since ap^ζp(ά), we have
From (2.4)
E
£c S» j (β I y I y» M(l+ I y I )2> e-w** dy
^c ε
β
" tf*"gc &* ζp(a) .
Moreover we have
E [ 1 8vG,(β1(ί)+flβ2(0)-9vG,(β1(ί)) I >; I Bβ) I >2a \ B2(t) \ ]
^C'£β"(\ (a\y\\x\*-\\+\x\γγ e-«'^W dxdy
J J \x\>2a\y\
^
c
'
£β
" \ ( J" ί (β"1)ί+1'"1 e~" dr) a(\y\γ e'^'2 dy,
for (l+r)2j> exp(— r2/2)^c exp(— r). Using the estimate
Ja\y\
We obtain the inequality
From (3.6) the proof is completed. q.e.d.
Next, we shall consider the moment inequality with respect to the process
ψ(α, X
r
). Let Gβ(x) be the function defined by (2.6).
Lemma 4.2. There is a constant C independent of 0</3<1 such that,
forQ<a<l,
(4.3) α
+sup E I G W)+<LB2(ί))-
924 T. KOMATSU AND K. TAKASHIMA
Proof. From (2.8) we see that
Immediately we have, for | v \ = 1, 2,
of Γ E I ffΌ^BΛt)) I p dt£c (aβ)* .
Jo
Using the estimate
/S I y I )* Γ ((log I x+θay \ )2+ | x+θay \ 3)> dθ
Jo
we have
Γ EIΘG^W+^WJ-SG^W)!^^^^Jo
It is much easier to show that
sup E\Gβ(B1(t)+aB2(t))-Gβ(B1(t))\^c^(aβ)p .
So the proof is completed. q.e.d.
In consideration of (2.9) we shall define, for | v \ =2,
(4.4) λ
v
(α, x,y) = ( \ (x+ay)* \ x+ay \ '2-^ | * | -2 1 + | log l
Lemma 4.3. Let />=4+8δ>4. There is a constant C such that, for
(4.5) E I Γ λ
v
(α, Bl(t)i B2(t))2 dt \ */2^ CJo
Proof. Divide the space R2* into three domains:
Z>(0, α)= {(*,y); |*|.V|j>|>-lρgα} ,
INTERSECTIONAL LOCAL TIME 925
and define
pk(<*> *y y) = JWα> (χ> y)
Let B(t)=^(Bl(t)9 B2(i)). First, we have
E\\\\*pQ)(a,B(t))dt\>'2Jo
£(E P λί'(β, B(t)) dt}W (E f 1
 A(β,Jo Jo
ί^. (E
 Po(a, B(t)) at)Jo
= c I log β|</*-1
 a
Since
Γ 15^ ) I -1 dt = -^- { IA(1) I - Γ 1 5ι(ί) I -1 BMJo a — 1 Jo
it holds that
E l f 1 \B1(t)\-1dt\9<oo foranyί>0.Jo
We see that, for any 0<μ,<l, using the mean value theorem,
(λj Pi) (a, x, y)
kl)2) (1+ kl2)}2 h(a, x,
£c (a -LzL)"(l+(logk|)4) (1+ l*l«) ft(β, *,
^.(βiogίi/β))^!*!-^!*!*).
Therefore we have, setting μ=(4+68)lp,
| Γ
Jo
Set r=(2+4δ)/S. Since (r-l)/>/r=4+6δ,
926 T. KOMATSU AND K. TAKASHIMA
E I ( ' (A* ft) («, B(t))dt\^
Jo
| Γ ft(β,
Jo
*• (α log(l/α))2+3δ (E I I A(0 1 ~l dt \ 4+6δ)1/2
These prove (4.5). q.e.d.
From (2.9) and (4.5) we know that
(4.6) ΣJ E| ja^fiiW+^WJ-β^fiiCO)!2 Λl"2^-^ «2(1+δ) .
Combining (3.5), (4.3) and (4.6), we obtain the following lemma.
Lemma 4.4. TV^tf w α constant C independent ofO<β<l such that
(4.7) E I .K2-2/3, -X;)-Ψ(2- A. -SΓT)— ψ<2-2/3,
^r, σ-^1, where />=4+8δ>4.
5. Proof of theorems
We shall prove Theorem 2 and 3 applying the following lemma. The
basic idea of the lemma is communicated by Prof. S. Kusuoka.
Lemma 5.1. Let {Φ
w
(τ)}, O^r^l, be a sequence of real valued continuous
processes. If there are positive constants C, p, q and δ mch that, for all r, σ and n,
(5.1)
then
(5.2) P [ Σ sup I φ
Λ
(τ)-Φ
Λ
(0) |< oo] = 1 .
Proof. Choose γ, 0<7<δ, and -ηy 0<^<1, so small that (1—97) (1+δ— γ)
1. Set
J(m) = {(ί,./)€=Z2+; 0^i<j^2myj-i
INTERSECT IONAL LOCAL TIME 927
Then, from (5.1) we know that
P [\Φn(j2-m)-Φ
n
(i2->»)\P>2-»«<2 ((j-iftΓ~γ for any (i,j)*=J(m)]
Let <4(M, JV) denote the set
A(M, N) = {\Φn(j2-m)-Φ
n
(t2-m)\*^2-»«'2 ((/-O 2- γ
for all n^N and (ί, j)^J(m) with m
Then we have P [^ (M, JV)] f 1 as M, JV f oo.
For a moment, we shall consider paths of processes {Φ
n
(τ); n^N} on
the set A(M9 N). Pick 0^σ<σ'^l so close that σ'—σ<2'M^-^. Choose
7W such that
and expand σ and σ' as follows:
σ =
σ
 =2--
where m<m(l)<m(2)< ••• and m<m'(l)<m'(2)< ••• . Since ΦM(τ) is con-
tinuous in r, we have
{2 Σ
Therefore
sup|ΦM(τ)-Φn(0)| ^c 2-nq/
This implies (5.2), for P [A(M, N)] f 1 as M, Λ' f oo. q.e.d.
Proof of Theorem 2. Let p>2 and 2— α=2yS>0. Then there is a posi-
tive constant 8 such that ^(a)^i:-a2(1+5), where ζp(a) is the function defined
by (4.1). Set S(n)=2~n and
From Lemma 4.1, the function Φ
Λ
(τ) satisfies condition (5.1) for q=βp From
928 T. KOMATSU AND K. TAKASHIMA
Lemma 5.1 we have
lim sup I Λ/r
ε(n)(α, Xτ)— ψ(a, Xτ)— ψ ,c«)(α, X^+^(a, X0) \ = 0 a.e. ,n+09 r
for
*τ) = Σ Φ*(τ)
Since i/r8(n)(#> J£"0)->ψ»(α, -X"0) w->oo, and since ιKω(α, ^r) is continuous in
T, we conclude that
P [ψ^α, XT) is continuous in r] = 1 . q.e.d.
Proof of Theorem 3 . Set
Φ.(τ) =
From Lemma 4.4, the function ΦM(τ) satisfies condition (5.1) for ρ=q=4+8S.
Therefore Ψ
n
(X
r
)— Ψ
n
(X0) converges uniformly in r as n->oo almost every-
where. By Theorem 1, ψ
n
(XQ) converges to Ψ(X0) a.e. as n-*oo. Hence
P [Ψ
n
(X
r
) converges uniformly in r as w— >oo] = 1 .
This implies that {Ψtt(w)} converges quasi-everywhere and
P [lim Ψ
n
(XT) is continuous in r] = 1 . q.e.d.
References
[1] M. Fukushima: Basic properties of Brownίan motion and a capacity on the Wiener
space, J. Math. Soc. Japan 31 (1984), 161-176.
[2] N. Ikeda and S. Watanabe: Stochastic differential equations and diffusion
processes, North Holland and Kodansha, 1981.
[3] T. Komatsu and K. Takashima: The Hausdorff dimension of quasi-all Brownian
paths, Osaka J. Math. 21 (1984), 613-619.
[4] N. Kόno: Proprietes quasi-partout de fonctions aleatoires gaussiennes, Lecture
Note of the University of Strasbourg.
[5] N. Kόno: ^-dimensional Brownian motion is recurrent with positive capacity,
Proc. Japan Acad. 60 (1984), 57-59.
[6] S. Kusuoka: Analytic functionals of Wiener process and absolute continuity,
Lecture Notes Math. 923, Springer-Verlag 1982, 1-46.
[7] I. Shigekawa: On the existence of the local time of the I-dimensional Brownian
motion in quasi everywhere, Osaka J. Math. 21 (1984), 621—627.
[8] K. Symanzik: Euclidian quantum field theory, in Local quantum theory (ed.
R. Jost), New York, Academic Press, 1969, 152-226.
INTERSECTIONAL LOCAL TIME 929
[9] S.J. Taylor: The Hausdorff a-dimensional measure of Brownian paths in n-space,
Proc. Cambridge Philos. Soc. 49 (1953), 31-39.
[10] M.J. Westwater: On Edwards9 model for long polymer chains, Comm. Math.
Phys. 72 (1980), 131-174.
[11] R.L. Wolpert: Wiener path intersections and local time, J, Funct. Anal. 30 (1978),
329-340.
Department of Mathematics
Osaka City University
Sugimoto 3-3, Sumiyoshi-ku
Osaka (558), Japan



http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-02104-15.pdf

On the existence of intersectional local time except on zero capacity set

Abstract

Similar works

Full text

Available Versions

Osaka University Knowledge Archive

Institutional Repositories DataBase (IRDB)

Osaka City University Repository